Poisson Errors: Estimating Typos Missed by Two Proofreaders

A text's total error count follows $\operatorname{Poisson}(\lambda)$. One motivating approximation uses 100 pages, 100 lines on each page, and 100 character positions per line: one million independent, identically distributed Bernoulli error opportunities. For a portion A with k character positions, the approximate mean error count is $\lambda k/10^6$. Two proofreaders independently inspect the text. Each error is detected independently by reader i with probability $p_i$, i=1,2. Define $X_1$ as errors found only by reader 1, $X_2$ as errors found only by reader 2, $X_3$ as errors found by both, and $X_4$ as errors found by neither. Estimate the distribution of $X_4$ and the benefit of an independent third reader who detects each error with probability $q$.

(a) Specify the joint law of $X_1$, $X_2$, $X_3$, $X_4$.

(b) Verify $E[X_1]/E[X_3]=(1-p_2)/p_2$ and $E[X_2]/E[X_3]=(1-p_1)/p_1$.

(c) Estimate $p_1$, $p_2$, $\lambda$ by replacing $E[X_i]$ with the observed X_i.

(d) Estimate the mean number of errors missed by both readers.

(e) With $X_1=60$, $X_2=30$, $X_3=40$, estimate the distribution of $X_4$.

(f) For these counts, an independent third reader has $q$=0.9. By how much does the expected missed-error count decrease?